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Factorials

Definition

A factorial of a positive number n is defined to be the product of all positive whole numbers less than and equal to n. E.g. n=5 5!=5×4×3×2×1. Note: We define the factorial 0!=1.

Worked Examples

Example 1

Calculate 7!.

Solution

From the definition we have 7!=7×6×5×4×3×2×1. Hence 7!=5040.

Example 2

Simplify 9!6!.

Solution

Firstly we can expand the factorials using the definition 9!6!=9×8×7×6×5×4×3×2×16×5×4×3×2×1. This can then be simplified by canceling terms on the top and bottom to give 9!6!=9×8×7. Therefore, 9!6!=504.

For more information see simplifying fractions.

Example 3

Simplify 16!13!8!

Solution

As above start by expanding the factorials 16!13!8!=16×15××3×2×1(13×12×11××3×2×1)(8×7×6×5×4×3×2×1). Then cancel out any terms that appear on the top and the bottom of the fraction 16!13!8!=16×15×148×7×6×5×4×3×2. This can be canceled further if we factorize the numerator 16!13!8!=8×2×5×3×2×78×7×6×5×4×3×2. Then finally we get 16!13!8!=14×3,=112.

Example 4

Simplify (n1)!(n+1)!.

Solution

(n1)!(n+1)!=(n1)×(n2)××2×1(n+1)×n×(n1)××2×1,=1(n+1)×n,=1n2+n.

See Also

The binomial expansion uses factorials to calculate the coefficients.

Test Yourself

Try our Numbas test on factorials.